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Show this may be simplified to The vorticity at the inlet plane will then have axial and tangential components, detennined by the radial gradients in Vz and va . Figures 14 and 15 show the axial and tangential velocity profiles along with the components of vorticity calculated from these profiles. In calculating the vorticity, the timemean value of each component has been fit with a cubic spline (the solid line) which was then differentiated to obtain the vorticity. It should be emphasized that the components of vorticity calculated here are only approximate since we have neglected not only the streamwise velocity gradients but also the fluctuations in the inlet velocity as indicated by the dashed lines in the figures. Nevertheless, these calculations give us a reasonable estimate of how the components of vorticity are distributed over the entrance. The combination of Figs. 14 and 15 show that the vortex lines3 entering the furnace take the form of left-handed helixes with the vorticity vector pointing in the downstream direction (+z,-9). In addition to the bulk vorticity, the discontinuity in velocity at the outer edge of the throat requires that there be a vortex sheet (with strength equal to the magnitude of the velocity jump) sourced from that edge. The vortex lines within this sheet also take the form of left-handed helixes but with the opposite sign as the bulk fluid, that is, the vorticity vector is again tangent to helical lines but points in the upstream direction of the flow (-z). Since there is no discontinuity in velocity at the surface of the centerbody, there is no internal vortex sheet. Note that since there is no internal vortex sheet, Stoke's theorem requires that the axial component of vorticity associated with the outer sheet be equal to the integral over the inlet plane of the axial component of vorticity in the bulk fluid. Likewise, the tangential component of the vorticity in the sheet must be equal in magnitude to the integral of the tangential vorticity of the bulk fluid over a transverse plane (one normal to the tangential direction) in the annulus. We can then think of the inlet flow as being composed of fluid with left-handed helical vortex lines of two signs. The outer fluid (vortex sheet) has concentrated vorticity in the -z, +9, direction along the helix, while the inner fluid (bulk) has its vorticity dispersed and in the +z, -9, direction along the helix. In the discussion which follows we will refer to these two portions of fluid as having negative and positive helical vorticity, respectively. We begin construction of the flow by simply extending the helical vortex lines from the inlet plane into the furnace in a manner consistent with the data presented earlier. Figure 16 shows the positive and negative helical vorticity extended into the flow. The vorticity associated with the bulk fluid is depicted by two vortex lines originating from the annulus at the inlet plane, 3 A vortex line is a line in the fluid whose tangent is everywhere parallel to the local vorticity vector. This is not to be confused with a line vortex, see Batchelor (1967). - 6 - |
